# All Questions

113,706
questions

**3**

votes

**1**answer

111 views

### Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...

**0**

votes

**0**answers

11 views

### Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...

**2**

votes

**1**answer

137 views

+50

### Relation between coefficients of expansions

Related to Relations between coefficients of expansions of a rational function at 0 and infinity
I commented at the linked question that the question seemed less about what happened "at infinity", ...

**0**

votes

**0**answers

5 views

### Radon transform range theorem and radial functions

In dimension 2, the Radon transform range theorem states that a function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if for all integers ...

**0**

votes

**0**answers

20 views

### Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...

**6**

votes

**1**answer

363 views

### Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...

**4**

votes

**1**answer

165 views

### Conjectured primality test for specific class of $N=k \cdot 6^n+1$

Can you provide a proof or a counterexample for the claim given below?
Inspired by Theorem 5 in this paper I have formulated the following claim:
Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\...

**1**

vote

**1**answer

51 views

### Sampling i.i.d. variables with restrictions

General Problem: Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "...

**4**

votes

**0**answers

54 views

### Automorphisms of $\mathbb{C}[x, y, z]$ over $\mathbb C[x]$

What are the automorphisms of $\mathbb{C}[x, y, z]$ fixing $\mathbb{C}[x]$? I.e. those automorphisms $\phi:\mathbb{C}[x, y, z]\to\mathbb{C}[x, y, z]$ s.t. $\phi(x) = x$. I am interested in complete ...

**2**

votes

**0**answers

26 views

### Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...

**2**

votes

**0**answers

33 views

### Sections of non-reduced schemes

Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...

**0**

votes

**1**answer

25 views

### Distribution of the direction of Gaussian random variable

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...

**2**

votes

**1**answer

78 views

### On a particular proof of “if the sharp of every real exists and every club contains a club constructible from a real, then $\delta^1_2 = \omega_2$”

I am referring to the proof of (4) implies (1) in Theorem 3.16 of Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. His proof leverages on the fact that if the sharp of ...

**1**

vote

**0**answers

9 views

### How do solutions to this quadratic congruence distribute as the number of factors grows?

This is a question that arose during a conversation with a colleague regarding Landau's fourth problem, which asks whether there are infinitely many primes of the form $n^2+1$. The Conjectured ...

**6**

votes

**1**answer

198 views

+50

### Boundedness of total current in electrical network (Banded graph)

Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...

**4**

votes

**1**answer

308 views

### Which books should I read in order to be prepared to study information geometry?

At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...

**6**

votes

**2**answers

276 views

### Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...

**2**

votes

**0**answers

33 views

### Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...

**2**

votes

**0**answers

76 views

### Size issues (small/large categories) when defining stacks in the Algebraic/differentiable/topological setting

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:
We will not distinguish between ...

**7**

votes

**1**answer

220 views

### Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...

**-2**

votes

**0**answers

80 views

### Asymptotically vanishing probabilities

Let $P_n$ be a sequence of probability measures on $(\mathbb{R}^{n},\mathcal{R}^{n})$, where $\mathcal{R}^{n}$ is the Borel $\sigma$-field associated to $\mathbb{R}^n$. For any sequence of measurable ...

**-2**

votes

**1**answer

164 views

### Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological ...

**-2**

votes

**0**answers

38 views

### About the Notation $\int |d\mu|$ [migrated]

I see the notation $\int |d\mu|$ lots of times and I could not find the definition of this notation. I can guess what it means, but I want to know the mathematically precise meaning. Thank you!

**0**

votes

**1**answer

29 views

### How to combine global standard deviation given several sample statistics?

Is there any approximation formula (best guess?) to calculate global std given multiple set statistics (size, mean, std)?
I have an aggregated statistics from several sets.
...

**0**

votes

**1**answer

255 views

### Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...

**7**

votes

**0**answers

115 views

### Can a covering space of the $p$-adic disc split over the circle?

Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$.
Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...

**1**

vote

**0**answers

21 views

### Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that
$\{b_1,\ldots,b_n \}$ is a ...

**1**

vote

**0**answers

65 views

### G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...

**2**

votes

**0**answers

130 views

+50

### The first part of the Hilbert sixteenth problem for elliptic polynomials

A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.
Inspired by the first part of the Hilbert ...

**0**

votes

**1**answer

31 views

### An alternative description of normalized cochains in terms of tensor powers of the augmented ideal

I want to know if the following alternative of the normalized non-homogeneous cochains is already know.
Let $G$ be a group and let ${\mathcal I}={\mathcal I}_G$ be its augmentation ideal, ${\mathcal ...

**3**

votes

**3**answers

302 views

### Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$

I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...

**-3**

votes

**0**answers

68 views

### $x^4=−1 \pmod p$ implies $p = 1 \pmod 8$ [closed]

Let $p$ be an odd prime. Show that
$$x^4=−1 \pmod p$$
has a solution if and only if $⇔$ $p=1 \pmod 8$
Can someone help me with this, please? I really want to understand the steps.

**-1**

votes

**0**answers

79 views

### The space of harmonic functions on an open set is infinite dimensional? [closed]

I want to prove that he space of harmonic functions on an open set $\Omega \subset \mathbb{R}^n $ , with $n \geq 2$, is uncountablely infinite-dimensional.
I guess that I have to find a linearly ...

**1**

vote

**0**answers

21 views

### Number of solutions to linear diophantine equations, with natural coefficients in a box

Let c, k, d $ \in \mathbb{N} $, let a, x $ \in \mathbb{N}^k $ suppose for all i $ \leq $ k, $ x_i \leq d $, $ a_i \in \mathcal{O}(d2^i) $ and $ \sum{a_ix_i} = c $ my question is for the value of c ...

**4**

votes

**0**answers

42 views

### What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...

**7**

votes

**0**answers

147 views

### Checking if a multiplication table represents a group [duplicate]

Given an $n \times n$ multiplication table,
can one check if it represents a group in $o(n^3)$ time?
All properties can be checked by mindless try-all possibilities loops:
Whether there is an ...

**1**

vote

**2**answers

313 views

### Two-dimensional Perron formula

There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series:
$$ \sum_{n\le x} f(n) = {1\over 2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) x^...

**0**

votes

**0**answers

327 views

### Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...

**4**

votes

**1**answer

70 views

### Rates of convergence to Tracy-Widom?

$\renewcommand{\!}{\mathbf}
\renewcommand{\Ai}{\operatorname{Ai}}$
One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where
$$\mathbf A(x, y)=\...

**4**

votes

**1**answer

126 views

+50

### Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...

**1**

vote

**0**answers

88 views

### Varieties with everywhere good reduction isomorphic over every completion

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the localization of $R$ at $\mathfrak{m}$ ...

**1**

vote

**0**answers

66 views

### Surjectivity of multiplicative map

Let $S$ be a smooth complex algebraic surface, and $\mathcal{F}$ be a coherent sheaf on $S$.
I want to consider $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{F} \boxtimes \mathcal{F}...

**4**

votes

**0**answers

104 views

+50

### Batalin-Vilkovisky integral is invariant under infinitesimal deformation

This is the basic theorem of Batalin-Vilkovisky integral, and these are three main versions of the theorem.
Articles about BV formalism almost certainly cite one of them, especially Schwarz's.
However,...

**0**

votes

**0**answers

31 views

### How prove this combinatorial-identities

if $n\ge 1$ be postive integers,and $x,y,z$ be any real numbers,such $x+y+z=n-1$,for any real number $a$.show that
$$\dfrac{(-4)^n}{\binom{2x}{n}}\sum_{r+s=n,r,s\in Z}\dfrac{\binom{y}{r}\binom{y-a}{r}...

**2**

votes

**1**answer

94 views

### Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...

**19**

votes

**5**answers

695 views

### Computation of fraction field of formal series over the integers

What is the fraction field $K$ of the domain $\mathbb Z[[X]]$?
It is strictly smaller than the field of Laurent series $L=\operatorname {Frac}\mathbb Q[[X]]$, since $\sum_{i\geq 0}\frac {X^i}{i!}\in ...

**2**

votes

**0**answers

24 views

### Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...

**2**

votes

**2**answers

73 views

### Finding the nearest quadratic Bézier curve

Given a set of three-dimensional quadratic Bézier curves.
I'm looking for some analytical solution to find the nearest curve to an arbitrary point in space.
Example
I already have a brute force ...

**4**

votes

**0**answers

88 views

### The Jacobson radical as a bimodule

Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...

**4**

votes

**1**answer

344 views

### An open problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$ It is an open problem and I did not find any ...